Welcome to Problem Solving: Finding the Needle in the Haystack

In the Thinking Skills syllabus, the section on Searching, scheduling and optimising is all about being efficient. This specific chapter, Search for solutions in simple situations, focuses on how we find an answer when we are given a set of rules and some data, but the answer isn't immediately obvious. Think of it like a puzzle where you have all the pieces, but you need a strategy to put them together.

Don’t worry if this seems a bit abstract at first. In simple situations, the "search" is usually just a matter of checking a few possibilities to see which one works. You don’t need complex math—just a clear head and a bit of organization!

1. What is a "Simple Situation"?

In the context of Paper 1 and Paper 3, a simple situation usually means:

  • There are only a few possible answers to check.
  • The rules (or "constraints") are easy to follow.
  • You only need basic arithmetic (\(+\), \(-\), \(\times\), \(\div\)) to test your ideas.

Note: If the problem involves many moving parts or requires you to find the "best" or "cheapest" option out of many, it might be an "optimal solution" or a "complex situation." We will cover those in other chapters!

2. The Three-Step Search Strategy

When you are asked to search for a solution, follow these three steps to stay on track:

Step A: Identify the Goal

What exactly are you looking for? Is it a time? A price? A person’s name? Identify and select relevant information (Syllabus 1.2) by ignoring the "fluff" in the story and focusing on the numbers or rules that matter.

Step B: Use a Systematic Search

A systematic search means checking possibilities in a logical order (like 1, 2, 3...) rather than jumping around randomly. This ensures you don't miss the correct answer.

Step C: Test Against the Rules

Once you find a potential answer, check it against the criteria. If the rule says "The total cost must be under \( \$50 \)" and your answer results in \( \$51 \), you must keep searching!

3. Common Search Methods

Depending on the question, you might use one of these two methods:

Method 1: Trial and Error (Trial and Improvement)

This is where you pick a starting point, see if it works, and then adjust your next guess based on the result.
Example: You are told a bus ticket costs \( \$3 \) for adults and \( \$2 \) for children. A group of 10 people paid \( \$24 \). How many adults were there?

  • Try 5 adults: \( (5 \times \$3) + (5 \times \$2) = \$15 + \$10 = \$25 \). (Too high!)
  • Try 4 adults: \( (4 \times \$3) + (6 \times \$2) = \$12 + \$12 = \$24 \). (Correct!)

Method 2: Exhaustive Search

This sounds tiring, but it just means "checking every single possibility." This is best when there are only 4 or 5 options. If a question asks which day of the week something happened, you only have 7 options to check!

4. Working with Simple Models

The syllabus mentions applying a simple model (Syllabus 2.2). A "model" is just a set of rules that describes a real-life situation.
Analogy: A mobile phone plan is a model. If the rule is "\( \$10 \) per month plus \( \$0.10 \) per minute," the "search" might be finding out how many minutes you used if your bill was \( \$15 \).

Quick Calculation:
Total Bill = \( \$15 \)
Fixed Cost = \( \$10 \)
Remaining for minutes = \( \$15 - \$10 = \$5 \)
Number of minutes = \( \$5 \div \$0.10 = 50 \) minutes.

5. Identifying Logical Relationships

Sometimes the "search" isn't about numbers, but about logical relationships (Syllabus 3.1). You might be looking for a pattern.
Did you know? Many "search" problems are actually just "elimination" problems. If you have four suspects and three of them have an alibi, the one left over is your solution!

6. Tips for Success and Common Mistakes

  • Don't Panic: If the first number you try doesn't work, it's not a failure! It's just narrowed down your search.
  • Read the Units: Always check if the question is in minutes vs. hours or cents vs. dollars. (The 2028 syllabus explicitly asks you to label your units!)
  • Show Your Working: Especially in Paper 3, credit is often awarded for "correct steps towards a solution" even if your final search result is slightly off.
  • Avoid the "Guess Trap": Don't just pick a number that "looks right." Briefly test it using the arithmetic from the scenario.

Quick Review: Key Takeaways

1. Selection: Only use the data that helps you solve the specific problem.
2. System: Check possibilities in order so you don't repeat yourself or miss the answer.
3. Verification: Always double-check your "solution" against the original rules of the problem.

Ready for more? Once you've mastered these simple searches, you can move on to "Search for solutions in more complicated situations" or learn how to "Find optimal solutions" where we look for the absolute best result!